Standard Form

Standard form is a short way to write very large and very small numbers. For example, the distance from the Earth to the Sun is about 150\,000\,000 km. In standard form this is 1.5 \times 10^8 km.

A number in standard form is written as two parts multiplied together:

A \times 10^n

A is a number from 1 up to, but not including, 10, so 1 \le A < 10. The power n is a whole number: positive, negative or zero.

So 4.5 \times 10^3 is in standard form. 45 \times 10^2 is not, because 45 is bigger than 10. 0.45 \times 10^4 is not either, because 0.45 is less than 1. All three are equal to 4500

In the exam you are usually asked to write a number in standard form, or to write a number in standard form as an ordinary number, for 1 mark each. Later questions ask you to put numbers in order of size, or to add, multiply or divide numbers in standard form and give your answer in standard form, for 2 or 3 marks. Some of these are on the non-calculator paper, so learn to do them by hand.

10^3 means 10 \times 10 \times 10, which is 1000. The power tells you how many 10s are multiplied together. It is also the number of zeros after the 1.

• 10^1 = 10 and 10^2 = 100
• 10^6 = 1\,000\,000, which is one million.
• 10^0 = 1

A negative power gives a number less than 1. Each time the power goes down by 1, you divide by 10. So 10^{-1} = 0.1, 10^{-2} = 0.01 and 10^{-3} = 0.001

Multiplying by 10^5 moves every digit 5 places to the left, so the number gets bigger. Multiplying by 10^{-5} is the same as dividing by 100\,000: every digit moves 5 places to the right, so the number gets smaller.

First find A: put the decimal point straight after the first digit that is not zero. For 36\,400 this gives 3.64, and for 0.0052 it gives 5.2

Then count how many places the digits move to get from A back to the original number. That number of places is the size of the power.

• A number of 10 or more has a positive power: 36\,400 = 3.64 \times 10^4
• A number less than 1 has a negative power: 0.0052 = 5.2 \times 10^{-3}
• A number from 1 up to, but not including, 10 has power 0: 7.5 = 7.5 \times 10^0

Keep any zeros that sit between other digits. 40\,700 is 4.07 \times 10^4, not 4.7 \times 10^4

When numbers are in standard form, compare the powers of 10 first. The bigger the power, the bigger the number. So 1.2 \times 10^5 is bigger than 9.8 \times 10^4, because 120\,000 is bigger than 98\,000

Take care with negative powers: -3 is bigger than -5. So 2 \times 10^{-3}, which is 0.002, is bigger than 8 \times 10^{-5}, which is 0.00008

If two numbers have the same power, compare the A parts. 6.1 \times 10^3 is bigger than 5.9 \times 10^3

This only works when every number is in standard form. If one is not, write it in standard form first, or write all the numbers as ordinary numbers.

You cannot just add the A parts and the powers. 3 \times 10^4 + 2 \times 10^3 is not 5 \times 10^7, and it is not 5 \times 10^4 either.

The safest way is to write both numbers as ordinary numbers, add or subtract, then write the answer in standard form: 30\,000 + 2000 = 32\,000 = 3.2 \times 10^4

If the powers are the same, you can add the A parts and keep the power. Then check that the answer is still in standard form. For example, 7.5 \times 10^6 + 4.5 \times 10^6 = 12 \times 10^6, which is 1.2 \times 10^7 in standard form.

To multiply, multiply the A parts, then add the powers:

10^m \times 10^n = 10^{m+n}

For example, (2 \times 10^3) \times (4 \times 10^5) = 8 \times 10^8

To divide, divide the A parts, then subtract the powers:

10^m \div 10^n = 10^{m-n}

For example, (9 \times 10^7) \div (3 \times 10^2) = 3 \times 10^5

Your answer may not be in standard form yet. If the first part is 10 or more, divide it by 10 and add 1 to the power: 35 \times 10^4 = 3.5 \times 10^5. If it is less than 1, multiply it by 10 and take 1 off the power: 0.4 \times 10^6 = 4 \times 10^5

On the calculator papers you can type a number in standard form with the \times 10^x key. If the display does not show standard form, write the answer in standard form yourself.

• Moving the digits too far. 3.4 \times 10^2 is 340, not 3400: the digits move only 2 places, so just one zero is needed.
• Getting the sign of the power wrong. A negative power makes the number small, so 3.6 \times 10^{-3} is 0.0036, not 3600
• Adding the A parts when the powers are different. Write both numbers as ordinary numbers, then add.
• Leaving the first part outside 1 to 10. 36 \times 10^{-4} is not standard form; it is 3.6 \times 10^{-3}, because a smaller first part needs a bigger power.
• Giving the final answer as an ordinary number. When the question says "Give your answer in standard form", change your final answer back into standard form.
• Comparing powers before every number is in standard form. 45 \times 10^2 is bigger than 3.1 \times 10^3, so write each number in standard form first.